The present invention relates to a tuning fork type quartz crystal resonator using a coupling between the fundamental flexural and fundamental torsional modes of vibration, and more particularly to the shape, size and cutting angle of the quartz crystal resonator.
Recently, a tuning fork type quartz crystal resonator having a favorable favorable frequency-temperature characteristic (hereinafter referred to as f-T characteristic) by utilizing the coupling between the flexural and torsional modes has received much attention. This kind of resonator has a very excellent f-T characteristic compared with ordinary resonators which are used for watches at the present time. The accuracy of ordinary watches is a monthly rate of less than about 10 seconds or less than about 15 seconds. On the other hand, the accuracy of watches having a resonator using coupling between the flexural and torsional modes may be yearly rate of less than 10 seconds.
In order to utilize the coupling between the flexural and torsional modes of the tuning fork type quartz crystal resonator, a method utilizing the coupling between the secondary flexural mode (hereinafter referred to as F2) and the fundamental torsional mode (hereinafter referred to as T1) has been adopted. According to this method utilizing F2, it is possible to minimize the R.sub.1 of the resonator. An example of such resonator utilizing the coupling between F2 and T1 is shown in U.S. Pat. No. 4,377,765.
However, by utilizing the coupling between the fundamental flexural mode (hereinafter referred to as F1) and T1, the size and the volume of the resonator may be smaller than those of the resonator utilizing the coupling between F2 and T1 and vibrating with the same frequency. Utilizing the coupling between F1 and T1 is advantageous for minimization of resonator.
The vibrating displacements of F1 and F2 are respectively shown in FIG. 1 and FIG. 2 by broken lines.
The reason why the reasonator utilizing F1 may be small is explained as follows. The frequency f.sub.F of the flexural mode (hereinafter referred to as f.sub.F) and the frequency f.sub.T of the torsional mode (hereinafter referred to as f.sub.T) may be expressed as follows: EQU f.sub.F =C.sub.F .multidot.w/l.sup.2 ( 1) EQU f.sub.T =C.sub.T .multidot.t/(lw) (2)
where; C.sub.F and C.sub.T are constants, respectively, t is the thickness of the resonator, and l and w are the length and width of the tuning fork arms respectively as shown in FIG. 3.
In the case of two tuning fork type quartz crystal resonators having the same shapes, the frequency of F2 is about 6 times the frequency of F1. Therefore, in order to obtain a fundamental flexural mode with a frequency equal to that of a secondary flexural mode, it is necessary to make w larger or to make l smaller in comparison with that of the secondary flexural mode resonator. In making w larger, it is difficult to take the reasonator to fit in the limited space of a housing. Therefore, making l smaller is the method generally used. In this case, so as to obtain F1 with a frequency equal to that of F2, the length of the tuning fork should equal to about 40% of that in case of the secondary flexural mode.
In order to utilize the coupling between the flexural and torsional modes, it is necessary to bring f.sub.T close to f.sub.F. The condition relative to f.sub.F and f.sub.T for the resonator having an excellent f-T characteristic which may be used for a yearly-rate-accuracy watch may be practically expressed as follows: EQU (f.sub.F -f.sub.T)/f.sub.F =0.02.about.0.05 (3)
Therefore, in the case of utilizing the coupling between F1 and T1, so as to satisfy the equation (3), it is necessary to make l smaller in comparison with the case of utilizing the coupling between F2 and T1, so that the thickness t must be made smaller. Consequently by utilizing the coupling between F1 and T1, the volume of the resonator is extremely smaller than that in the case of utilizing the coupling between F2 and T1.
If the tuning fork type quartz crystal resonator is produced by photolithography, to make the resonator thin results in a shortening of the etching time for the quartz crystal. As described above, utilizing the coupling between F1 and T1 minimizes the size of the resonator. Furthermore, it is possible to produce the resonator in a short time by utilizing photolithography, so that it is advantageous for reducing the manufacturing cost.
Utilizing the coupling between F1 and T1, as the length l of the tuning fork becomes shorter, the whole length of the resonator becomes shorter too. Therefore, the vibrating displacements of the flexural and torsional modes in the end region of the base which is the portion for supporting the resonator are apt to enlarge. The term "base" refers to the portion of the resonator except the two tuning fork arms. The large vibrating displacement causes a vibration leakage by the vibration of the resonator 41 transmitted to a housing 44 through lead terminals 42 and a plug 43 shown in FIG. 4. If the vibration leakage is large, the Q value of the resonator becomes lower and the R.sub.1 becomes higher, so that a stable frequency-temperature characteristic or a stable frequency-aging characteristic may not be obtained.
Conventionally, in the case of the tuning fork type quartz crystal resonator utilizing the coupling between the flexural and torsional modes, it has been found that the resonator having a cutting angle effective to get an excellent f-T characteristic which may be used for a yearly-rate-accuracy watch may be obtained by rotating the Z plane in the range of -15.degree. to 0.degree. with the counterclockwise direction of rotation around the electrical axis (x axis) of the quartz crystal being defined as a positive angle, where the Z plane means the plane perpendicular to the optical axis (z axis) of the quartz crystal. In the case of utilizing the coupling between F1 and T1, however, and of obtaining a very small sized resonator having an excellent f-T characteristic for a wristwatch, we found that it is undesirable to use the conventional cutting angle.